The present invention provides rapid and accurate reconstruction of acoustic radiation from an arbitrarily shaped vibrating object.
Known methodologies deal with reconstructing acoustic radiation from a vibrating structure, also known as acoustic holography, which involves determining the vibration patterns on the surface of a structure based on simple acoustic pressure measurements from an array of microphones near the structure. A brief comparison of various methodologies that have been developed over the past two decades is given below.
The original nearfield acoustic holography (NAH) uses a plane wave expansion to represent the radiated acoustic field. The advantage of a planar NAH is that if the acoustic pressures over an infinite plane (called holography plane) can be measured continuously and exactly, the acoustic pressure and the normal component of the particle velocity anywhere including the source surface can be reconstructed with an infinite resolution. However, such a scenario can never be realized because in practice measurements are always discontinuous, measurement area is finite, and dynamic range of measurement devices is limited. Moreover, measurement and reconstruction are confined to a planar surface. Even so, this approach is not applicable when there are reflecting surfaces near the source or when the source is confined to interior space. These restrictions make the planar NAH suitable for reconstruction of acoustic radiation from a planar source, but not for an arbitrary surface in an exterior region.
To reconstruct acoustic radiation on the surface of an arbitrary object, one can utilize the Helmholtz integral theory that relates the field acoustic pressure to the acoustic quantities on the surface of the object. While this integral formulation is advantageous for a general surface, it has several inherent drawbacks.
The first drawback is the nonuniqueness of solutions to the surface Helmholtz integral equation. While this drawback can be overcome by the CHIEF method, there is no systematic way of determining the over-determined points in the interior region to yield a unique solution. The second drawback lies in the fact that the acoustic field is determined via a spatial discretization. In other words, the acoustic quantities are specified on certain discrete nodes and the measurements must be taken over a surface that completely encloses the source at once. For an arbitrary structure such as an engine vibrating in the low- to mid-frequency regime, the total number of discretized nodes on the surface can be very large, which necessitates taking a large number of measurements around the source. While one can use an iteration scheme to select the optimal measurement locations that may lead to more accurate reconstruction and that may reduce the overall number of measurement slightly, the process of this scheme itself can be very time-consuming. In any event, this approach requires the knowledge of source geometry and dimensions so as to generate a surface mesh. For an arbitrarily shaped structure, such a process can be very complex and time-consuming.
Another technique known as the HELS method is described in U.S. Pat. No. 5,712,805. In this method, the radiated acoustic pressure is expressed as an expansion of basis functions. The coefficients associated with these basis functions are determined by matching the assumed solution to the measured acoustic pressures. The errors incurred in this process are minimized by the least-squares method. Note that since the problem is often ill posed, it is critical to determine an optimal number of expansion functions, which depends on the signal to noise ratio (SNR), dynamic range of measurement devices, and standoff distances. The higher the SNR and dynamic range and the smaller the standoff distance, the larger the value of the optimum expansion number and the higher the accuracy of reconstruction.
The HELS method thus developed has been used to reconstruct acoustic radiation in both exterior and interior regions. One unique feature of this method is that it imposes no restrictions on the measurement locations. Moreover, the measurement aperture can be set comparable to the reconstruction area and reconstruction on the source surface can be carried out on a piecewise basis, which makes the whole process flexible and versatile.
It must be pointed out that the principle (i.e., expansion theory) underlying the HELS method has been discussed extensively in the past. For example, it has been used to analyze directivity patterns, far-field acoustic radiation based on nearfield measurements, sound radiation from a violin and antenna, noise source, and nearfield acoustic scanning.
An alternative to the spherical expansions is a collocation method first presented by Frazer, et al. as a means for satisfying differential equations at discrete points with a series expansion, which satisfies boundary conditions exactly. Alternatively, one can write the acoustic pressure in terms of a series expansion that satisfies the Helmholtz equation, whose coefficients are determined by requiring the solutions to satisfy the boundary conditions at certain discrete points in the least-squares sense. This finite-series expansion solution is discussed by Collatz and used by Meggs and Butler to predict far-field radiation based on the nearfield measurements.
Note that this boundary-collocation method is a variation of many related numerical techniques used to solve partial differential equations. These techniques are based on an assumed solution that satisfies the equation and/or boundary conditions exactly. The coefficients associated with this assumed solution can be determined using the least-squares, subdomain, collocation, or Galerkin""s method, which are special cases of the general criterion that the weighted averages of the residual error must vanish. Each of these methods yields a different set of weighting functions.
All these techniques require the assumed solution to satisfy the boundary conditions at a number of points equal to that of the expansion coefficients. These unknown coefficients are then determined by taking a direct or pseudo-inversion of the resulting matrix equation. Such an approach works for prediction of far-field acoustic radiation that can be described effectively by a few expansion functions. However, it cannot be used to reconstruct the acoustic field on the source surface. This is because the inverse acoustic radiation problem is ill posed. Hence any slight error in the input data may be so magnified in the inversion of the matrix that the reconstructed acoustic field can be completely distorted.
Unlike the spherical expansion, the HELS formulation provides an approximate solution for the entire exterior region, with relatively higher accuracy of reconstruction outside the minimum sphere that encloses the source under consideration, and relatively lower accuracy inside. Moreover, one has complete freedom in selecting the measurement locations, can set a measurement aperture equal to the size of reconstruction, and carry out a piece-wise reconstruction over the source surface. This is in contrast with the Helmholtz integral theory based NAH, which requires taking measurements over a control surface that completely encloses the source. The size of the matrix equation of the HELS method is also much less than that of Helmholtz integral formulation, hence the former it is computationally more efficient than the latter. However, the accuracy of reconstruction provided by the HELS method can be poor for an irregularly shaped surface. This is because the number of basis functions necessary for reproducing the acoustic field on a rough surface may be significantly increased, while the presence of measurement errors keeps the number of expansion functions down, thus lowering the accuracy of reconstruction.
The above review indicates that none of the methodologies are perfect. The planar NAH is convenient for a planar source, but its use is very limited. The Helmholtz integral theory based NAH is advantages for an arbitrarily shaped surface, but is not practical. The HELS method is practical, but the accuracy of reconstruction is limited for a general source of smooth surface. In addition, all these methodologies suffer from a common failing; namely, they are not applicable when there are a plurality of sources or reflecting surfaces in the vicinity of the source under consideration.
The new algorithm used in the method and apparatus of the present invention is called xe2x80x9cComplete Reconstruction of Acoustic Field Technologyxe2x80x9d or xe2x80x9cCRAFTxe2x80x9d, which combines the advantages of the HELS method and the Helmholtz integral theory based NAH, but does not require a prior knowledge of the source geometry and dimensions. Accordingly, reconstruction can be done based on simple acoustic pressures measured in the field only, which significantly improves the numerical computation efficiency. Moreover, this algorithm can be used to reconstruct acoustic radiation from a plurality of sources or from a single source surrounded by multiple reflecting surfaces.
The input to the CRAFT algorithm is the field acoustic pressures measured around a vibrating object under consideration. No prior knowledge of the geometry and dimensions of the source are required. Once the formulation is established, the entire acoustic field that includes the surface and field acoustic pressures, normal component of the surface velocity, normal component of the time-averaged acoustic intensity, and acoustic energy flow is completely determined. Moreover, this algorithm allows for separation of the nearfield acoustic quantities from the farfield ones, thus enabling one to identify the components of surface vibrations that are responsible for sound radiation into the surrounding fluid medium.